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104 V NORMED SPACES
2. Necessity. If u is continuous at the point (0,0), there exists a ball B:
sup (||xi||, I|x2ll) ^ r *n EI x E2 such that the relation (xi9 x2) e B implies ll«(*i» *2)ll <1- Let now (xl9 ;x2) be arbitrary; suppose first x± =£ 0, x2 ¥> 0; then if zt = rxj\\x^\\9 z2 = rx2/\\x2\\, we have HzJ = \\z2\\ = r, and there- fore ||M(2Ti,z2)II <1- But w(Zi, z2) = rX^i^2)/lkill 'il^L and therefore IW*i, *i)ll < a * ll*ill " II^2II with * = !/r2- If *i = 0 or *2 = 0, u(xl9 x2) = 0, hence the preceding inequality still holds. |
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(5.5.2) Let u be a continuous linear mapping of a Banach space E into a
Banach space F. If (xn) is a convergent (resp. absolutely convergent) series in E, (w(xn)) is a convergent (resp. absolutely convergent) series in F, and |
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The convergence of the series (w(xn)) and the relation £ u(xn) = w(£ ;*;„)
n n
follow at once from the definition of a continuous linear mapping (see
(3.13.4)). From (5.5.1) it follows that there is a constant a>0 such that II "CO II ^ a ' \\xn\\ f°r every n, hence the series (u(xn)) is absolutely convergent by (5.3.1) if the series (xrt) is absolutely convergent. |
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(5.5.3) Let E, F, G be three Banach spaces, u a continuous bilinear mapping
0/E x F into G. If(xn) is an absolutely convergent series in E, (yn) an absolutely convergent series in F, then the family (u(xm,yn)) is absolutely summable and |
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Using the criterion (5.3.4), we have to prove that for any /?, the sums
\\u(xm 9 yn)\\ are bounded. But from (5.5.1), there is an a > 0 such that
<p
, JV)II < all^mll ' \\yn\\, hence
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w = 0 / \n = 0
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which is bounded, due to the assumptions on (xn) and (yn). Moreover from
(5.3.6) and (5.5.2) it follows that, if s = £ xn , s' = |
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oo/oo \ oo
Z W(*m » ^n) = Z Z W(Xm , 3>n) I = Z W(X
m, n m = 0\n=0 / m = 0 |
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